Complex Analysis: Cauchy Integral Formula

In summary, the problem is asking for the value of the Cauchy Integral \int_{C} \frac{f(\zeta)}{\zeta-z} d\zeta when z is outside of the counterclockwise oriented circle C. The Cauchy Integral formula states that the integral is equal to zero if the function is holomorphic over the entire circle C. The question is seeking further insight or clarification on this topic.
  • #1
masterslave
8
0

Homework Statement


The problem, for reference, is from Sarason's book "Complex Function Theory, 2nd edition" and is on page 81, Exercise VII.5.1.

Let C be a counterclockwise oriented circle, and let f be a holomorphic function defined in an open set containing C and its interior. What is the value of the Cauchy Integral, [tex]\int_{C} \frac{f(\zeta)}{\zeta-z} d\zeta
[/tex], when z is in the exterior of C?

Homework Equations


The Cauchy Integral formula, as mentioned in the problem.


The Attempt at a Solution


I haven't the slightest how to begin the problem. My intuition, though, tells me [tex]\int_{C} \frac{f(\zeta)}{\zeta-z} d\zeta=0[/tex]. Any insight is appreciated.
 
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  • #2
If z is in the exterior of C then f(zeta)/(zeta-z) is holomorphic in zeta over all of C.
 

Related to Complex Analysis: Cauchy Integral Formula

1. What is the Cauchy Integral Formula?

The Cauchy Integral Formula is a fundamental theorem in complex analysis that relates the values of a holomorphic function inside a closed curve to the values of the function on the curve itself.

2. How is the Cauchy Integral Formula derived?

The Cauchy Integral Formula is derived from Cauchy's Integral Theorem, which states that the integral of a complex function over a closed path is equal to the sum of the function's values at all points inside the path. The Cauchy Integral Formula is then obtained by dividing this integral by 2πi.

3. What is the significance of the Cauchy Integral Formula?

The Cauchy Integral Formula has numerous applications in complex analysis, including evaluating integrals, calculating residues, and solving differential equations. It is also a fundamental tool in the study of conformal mappings and analytic continuation.

4. Can the Cauchy Integral Formula be extended to functions with singularities?

Yes, the Cauchy Integral Formula can be extended to functions with isolated singularities. In this case, the integral is calculated over a curve that encloses the singularity, and the integral is equal to 2πi times the function's residue at the singularity.

5. Is the Cauchy Integral Formula only applicable to complex functions?

Yes, the Cauchy Integral Formula is specific to the field of complex analysis and can only be applied to functions of a complex variable. It is not applicable to real-valued functions.

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